Approximately, with big error margins.

When historians compare different civilizations' wealth, they typically estimate a GDP per capita for that country for that time period. GDP is actually a measure of flows, not stocks, more akin to your annual income than your wealth. It has some desirable properties for economic historians: 

• household production of goods for own use (e.g. growing your own crops, building a house) counts as much as such production for other people. So if you find say the remains of a village longhouse, you don't need to worry about who built it. 

• household production of services for own use (e.g. cooking your own meals, or family prayers) doesn't count.  So you don't need estimates of how people spend their leisure time nor do you need to try to compare, say, the cultural value of English Morris dancing versus Northwestern American potlatches versus answering questions on r/askhistorians.  

• Depreciation of capital isn't included (this is the 'gross' bit of "gross domestic product"). So depreciation doesn't need to be estimated. 

• GDP measures what is produced within an area regardless of who owns that income - so you don't need to measure monetary flows.

• "per capita" means that you don't need to worry about how big the population was. 

The starting point is to assume a certain base level of GDP per capita on the basis that people on aggregate are producing enough to keep themselves alive and raise the next generation, then add on based on what evidence is available. If the archaeological evidence is that cities are getting bigger, then GDP per capita must be a bit higher, not only to build the city but to transport food, water, building supplies, etc to the people in said city, and take away waste. If there's evidence of increased trade, GDP per capita must be higher. 

Note that GDP isn't a measure of welfare, a group of nomads might well be taller, happier, better-fed and longer-lived than a village of farmers always skirting the edge of famine, but if the villagers have roads, walled fields, an irrigation system, a stone church, etc, it's reasonable to say that their GDP per capita was higher (but we might be missing something! what if the nomadic group had a tradition of complex, time-consuming textile production?) 

As you can imagine, this is all fairly arbitrary. The main collection of long-term economic statistics is the Maddison project, started by Angus Maddison, a British economist who was the historical professor at the Dutch University of Groningen, and continued after his death by his former colleagues and students. This project brings together estimates of historical GDP from multiple different sources, including the work of economic historians looking at specific areas for specific spans of time. The documentation for the latest update of the Maddison statistics (Bolt, Zanden, 2013) has four quality levels, starting with "official estimates of GDP, made by national statistics offices or by international agencies" and winding up with "guessestimates" (page 4). They state that subsistence level GDP is estimated to be between 250 and 300 per capita in 1990 US dollars (page 12). 

So, roughly, an economic historian estimating GDP per capita say for the Roman empire in 1 AD, is taking subsistence level as a starting point, then looking at written and archaeological evidence about economic activity: trade, mining, urbanisation, meat consumption, etc, and not just for 1AD itself but how these factors changed over time.  

On top of this, we often have records of price data from various sources, such as letters, or institutional accounts, on food costs and housing, and wages paid to employees. This allows estimates of real wages for labourers (though unfortunately data sources mostly don't state how many days a year people worked), which can be compared to the GDP per capita estimates. 

Estimates for different areas can also be roughly compared to each other for consistency. 

As you can see, the term "guesstimate" is well justified. Angus Maddison's original idea was to get some numbers down and that would provoke people into criticising them and trying to make a better estimate. It seems to be working.

Sources 

The Maddison Project: collaborative research on historical national accounts, Bolt, Jutta; Zanden, Jan Luiten. Economic History Review. Aug2014, Vol. 67 Issue 3, p627-651, working paper https://www.rug.nl/ggdc/historicaldevelopment/maddison/publications/wp4.pdf

See more papers on the Maddison Project at https://www.rug.nl/ggdc/historicaldevelopment/maddison/research.

United Nations, System of national accounts 2008, https://unstats.un.org/unsd/nationalaccount/sna2008.asp