My partner (who is an economic historian) and I did a quick back-of-the-envelope calculation on this.

There are 3 comparisons I can think of in which we can estimate how much 9,000 x 25,000 sesterces (225 million seterces) is worth today (in USD).

For wheat and labour, we looked at this paper by Peter Temin: The Economy of the Early Roman Empire.

These prices are all from the 1st century CE, with Didius Julianus paying the money in 193, but we're assuming prices were relatively stable. The paper mentions prices for wheat were relatively stable across 2 centuries.

Food How much food could you buy for 225 m sesterces, and how much US dollars is this food worth today?

According to Temin's paper,

For calibration, one modius (6.5 kilograms) of wheat cost four to six sesterces (on the private market) in Rome during the first century.

Let's call it 5 sesterces for easy calculations. Per kilo that works out to about .77 seterces. This means every soldier would be able to buy 32,468 kilos of wheat with their money. Let's hope none of them have a gluten intolerance...

Today, a metric ton (1000 kg) of grain costs $150, or $0.15 per kilo. The 32,468 kilos of wheat would only be worth $4870.13, and the total amount spent would be worth 43.8 million dollars.

Silver How much is the metal value of the coins, and what would that buy you today?

A sestertius is one quarter of a denarius. In 193, the denarius would have weighed 3.41 grams (80% pure), so the sestertius is a quarter of that - .85 grams, of which about 80% is silver^1, so .71 grams of silver. 25,000 sesterces, expressed as 6250 denarii, would be 17.050 kilos of silver.

Silver is worth $.53 per gram, so 25000 sesterces, just for their silver content, would be worth $9036.50, for a total of 81.3 million dollars for all 9000 praetores.

Labour How much money does a labourer get paid, and how does that compare to today?

during the first century CE, and the daily wage was between three and four sesterces.

We'll use a modern wage of $30,000 for comparison, and an average 1st century wage of 3.5 sesterces per day.

The sum paid to each praetorian is approximately 3,000 days' labour, or ballpark 10 years' wages for an ordinary worker.

If we say an ordinary unskilled worker earns maybe $30,000 USD/year, that's about $300,000 per labourer. That comes out to 2.7 billion dollars, measured in terms of purchasing unskilled labour.

^1 The Decline and Fall of the Roman Denarius, page 216, Figure 1.

Edits: Sources and calculation errors

Quick clarification: Are you interested in converting this to a modern dollar value in relation to commodity money (tying the sestertius to gold or silver and measuring the modern value of the quantity,) or are you interested in comparing it to other things that over 200 million sesterces could have bought you in 2nd century Rome?

Like you say, an exact currency conversion is impossible, but I think this is a very interesting question and I hope it gets an answer.

When asking what this would equate to in modern terms, I meant how much in dollars this would cost based on the gold/silver content of the coin and gold/silver prices today.

It's impossible to answer this question with precision, because the reign of Didius Julianus fell during a decline in the precious metal content of the physical coins. (PDF) The rate of decline varied regionally, and was often done covertly (with acidic surface-enrichment techniques used to ensure that the surface of the coin was relatively pure, while the core was heavily alloyed with base metals). Because the surviving examples of coins may not always be representative, the core testing methods are often destructive, and the coins in circulation may not all have been current, we can't be confident in giving exact metal values.

To even begin giving a broad estimate, let's first dispense with the units. A sestertius is a bronze coin with negligible metal value, but we're probably not speaking of physical sestertii. The sestertius was commonly used as a unit of account (much like a modern American is vastly more likely to speak of a thing being worth "ten dollars" than "a thousand cents") . Soldiers were conventionally paid in silver denarii, each of which was worth four sestertii in Imperial money. Therefore each soldier's haul would be 6250 denarii.

After the reforms of Augustus, each denarius weighs about 3.9 grams, making each payment about 24 kg of coins. In 193, those coins are approximately 60% silver, for a total of 14.625 kg of silver per man. As I'm writing this, the spot price of silver is just about US$525 per kilogram.

That makes a contemporary value of the payments (to beat the dead horse further, very approximately) US$7678 worth of silver per soldier, and US$69,103,125 in total.

On top of all the other qualifiers, please remember that these numbers can still be very misleading even if we could be confident in the fineness of the coins, because the buying power of a kilogram of silver in 193 was not the same as the buying power of the same kilogram today; calculating the specie value is really only useful in helping you visualize the physical size of the payments. If you'd like a good, readable overview of the difficulties inherent to converting prices from antiquity, and a survey of the available evidence of contemporary prices that can help you get a feel for how highly a Roman would have valued a given amount of money, consider checking out Sitta von Reden's Money in Classical Antiquity.