One thing to note: what makes Ukraine stand out from other former Soviet republics and other Eastern European countries is less that its GDP fell by half in the 1990s, but that it has had relatively anemic growth since then. Russian GDP also fell by roughly half in the 1990s, and didn't regain its 1990 level until around 2006. All former Soviet republics fell, especially in the early 1990s, and while Ukraine fell badly, it didn't fall the furthest (Azerbaijan and Georgia take that dubious honor). Most of the republics hit bottom around 1996 before beginning to regrow. Ukraine, and Moldova, however, are unique in that they haven't yet reached their 1990 GDP levels yet.

So why did these countries' economies implode? Essentially because a Union-wide production and distribution system completely broke down in the late Soviet period, at a time when macroeconomic instability massively increased. To put it succinctly, the republics, asserting their sovereignty even before the 1991 collapse, retained control of products, resources and revenues, preferring to barter with other republics or regions and starving the central government of revenue (a gap which was filled by printing lots of money, causing massive inflation). The Soviet government’s budget deficit in 1991 exceeded 20% of GDP, foreign loans exploded to $56.5 billion, and the economy had declined by 6% in 1990 and would decline by a further 17% in the first nine months of 1991. Inflation was running at 250%. The former Soviet republics didn't even fully disentangle their monetary systems until 1993, when Russia retired Soviet ruble notes and ended its connection with the former Soviet "ruble zone". All republics had major issues of state budgets spending vast amounts on subsidies to largely non-performing industries and having twin issues of inflation and economic decline.

In addition to the Union-wide, centrally planned economy coming apart, there was also an issue of demilitarization. Even in the last Gorbachev years, the size of the Soviet military and its expenditures were drastically reduced (the number of military personnel alone fell from 5.3 million servicemembers in 1985, to about 4 million in 1990, to about 1.7 million in 1994), and in an economy where an estimated 15-20% of GDP was spent on defense, this was a major shock. Yeltsin, coming on the heels of Gorbachev's defense cuts, in turn cut defense procurements by perhaps 90%. The idea (both in Gorbachev's time and Yelstin's) was that rapid demilitarization would allow industries to reorient towards consumer goods, but it's not easy to retool missile factories to produce televisions, especially in a state of political and macroeconomic chaos. Ukraine in particular was saddled with heavy industries that were either "rustbelt" industries (like coal mining or steel production) or heavily geared towards producing for a Soviet-wide defense industry, like naval shipyards.

One final point point - you might be interested in checking out this answer I wrote comparing the economies of Poland and Russia in the 1990s. All former Eastern Bloc states faced major economic downturns in the early 1990s as they dismantled state-run economies, but former Soviet states like Russia and Ukraine faced additional challenges in building new political and legal structures while also trying to build essentially new market economies from scratch.

Nevertheless, there is a case to be made that the 1990s declines in former Soviet states are somewhat exaggerated. Part of this is because any estimates of the size of the Soviet economy based on value are just that - very disputed estimates, as determining value added or the worth of capital goods didn't really translate to the systems used in market economies. Also, the collapse of the economy in post-Soviet states was as much a collapse of the official economy as anything - it didn't capture a vast black market in the lawless 1990s, nor did it capture well the semi-legal "gray economy", such as shuttle traders buying and selling goods in local bazaars. The Russian government itself estimated that the "shadow economy" was nearly 50% of GDP in 1996.

ETA - I'm sorry that this veered into a Russia-based answer and is less Ukraine-specific, but hopefully it provides some context. One major difference between the two countries of course is that Russia is a major natural gas and oil exporter, and Ukraine mostly relied on natural gas imports at below-market rates.

ETA #2: Some sources worth checking out are:

• Stephen Kotkin Armageddon Averted: The Soviet Collapse, 1970-200

• Gerald Easter. Capital, Coercion, and Postcommunist States

• Andrei Shleifer and Daniel Treisman. "A Normal Country: Russia After Communism". Journal of Economic Perspectives—Volume 19, Number 1—Winter 2005—Pages 151–174. PDF here.

• Serhii Plohky's The Gates of Europe: A History of Ukraine discusses some of the specific issues around economic transition in Ukraine in the 1990s in its later chapters.

To add to u/Kochevnik81's comment, here's a more Ukraine specific answer I had already drafted, though it covers much of the same ground.

It is very uncommon to achieve such a large collapse in GDP, so it may not be that surprising that there were multiple causes of this collapse. 

1. With the break up of the Sovet Union, the Ukrainian economy suffered a sharp fall in demand when existing commercial relations and trade patterns were broken up. This was common to all former Soviet Union (FSU) countries, and all of them saw output drops.

2. Structural reform in Ukraine was slower than in other FSU countries, reducing flexiblity to adjust to the external changes. For example, the Ukraine government suddenly found the price it paid for energy imports rising to world levels (before then, Ukraine is estimated to have been the biggest recipient of indirect transfers of energy resources from Russia in the FSU, (Dąbrowski 1995, page 10)), but these price prices were only passed onto enteprises in March 1995: before then the government was covering this huge cost itself. 

3. This lack of structural reform placed the Ukraninan government budget under massive pressure, and with limited ability to raise taxes, it resorted to inflation. This led to .... 

4. Hyperinflation, the World Bank estimated that inflation reached 10,000% in 1993. This hyperinflation was driven both by the government directly printing money and issuing credit and also by mandating that the financial sector extend huge amounts of nearly interest-free credits to  various large enterprises (Kravchuk, 1998, states that these reached some 16% of GDP in 1992). The level of hyperinflation was such that when a currency reform was carried out in 1996, the rate was 100,000 karbovantsi to 1 hryvna - i.e. lopping 5 digits off prices.

On top of this chain of events, there were other issues:

•  Ukraine generally maintained price controls on a large number of goods, with these only being lifted in 1996 (IMF, page 25) (there was a complex series of price adjustments, reforms, etc over 1992-1994, but generally prices were regulated one way or another). So hyperinflation plus price controls means that it was very hard for enterprises or households to find goods to purchase, and enterprises were resorting to barter.  Externally, the trade and payments system was only liberalised in the latter half of 1994 (page 48), so before then Ukrainians were limited in their ability to purchase abroad before that, again, enterprises would barter as part of trade, authorities esimtated that barter was more than 45% of exports in 1994 (IMF, 1997, page 48). 

• The Ukranian executive had weak controls on government expenditure, so a Ukrainian ministry could spend money that hadn't been approved. And during Soviet times, enterprises were subject to relatively strict controls in terms of spending, but in the early 1990s these were dropped, and the executives of enterprises regularly lobbied for and got subsidies, tax exemptions, etc. In other words, budget constraints were soft, worsening the budget problems.

• Ukraine's national government lacked capacity in other ways, e.g. the IMF assessed tax administrators as lacking resources in almost all reas, including the legal system not defining the rights of taxpayers versus the revenue department, accounting methodologies not meeting international standards, etc. As another illustration of the lack of capacity, local authorities often acted in contradiction to the national government, e.g. restricting grain exports.   

During this time, Kravchuk (1998) says that Ukraine gained a reputation as the "most economically illiterate nation on earth" (unfortunately he doesn't source this, I don't recall hearing this before myself, and a Google doesn't find any other references, though admittedly a lot less news commentary was online during the 1990s.) 

So, in short, the external situation of the economy had suddenly changed, making the old system unuseable. The government didn't reform in a way that allowed enterprises and households to adapt to the new system, instead it made things worse by running massive inflation instead. 

The fall in GDP may have been overstated, Ukrainians unsurprisingly responded to the hyperinflation and tax uncertainty and etc by engaging in informal economic activity.  The IMF noted that estimates of this informal activity ranged from about 30% to 100% of recorded GDP (IMF 1997, page 10). Properly, the informal sector should have been included in published GDP, including "household production of goods for own consumption" (the controversial exclusion from the definition of GDP is "household production of services for own consumption"), but it's hard to measure the informal economy at the best of times, and this wasn't the best of times for the Ukraine.

Sources

Kravchuk, Robert S. 1998. Budget Deficits, Hyperinflation, and Stabilization in Ukraine, 1991-96. Public Budgeting and Finance 18 (4): 45–70.

IMF, 1997, Ukraine, Recent Economic Developments, IMF Staff Country Report, 1997, https://www.imf.org/external/pubs/ft/scr/1997/cr97109.pdf

Marek Dąbrowski & Rafał Antczak,  1995, Economic Transition in Russia, the Ukraine and Belarus in Comparative Perspective, Center for Social & Economic Research https://case-research.eu/upload/publikacja_plik/3459873_050.pdf

Note that today you are supposed to believe that Ukraine started degrading since 1990s because of the "Russia intervention" in 2014: https://www.reddit.com/r/MapPorn/comments/hg9yn4/is_your_country_more_developed_than_ukraine_1990/fw2nk4x/

As native Ukrainian lived in an industrial town I can tell you that after communism disappearance the businessmen became new owners. Businessmen are not interested in GDP. They've just scraped most of factories.