I think the operative phrase in your post is that this involves stocks/options you "haven't been able to afford to exercise until now"
You're already acknowledging that affordability of a $5k stock investment is potentially an issue for you, even now, so I would advise proceeding with caution.
Since the company isn't publicly trading, there's no realistic way for you to sell the shares (you could sell them to someone privately if you could find them and arrange it), so it doesn't matter what their "fair market value" is. I think the valuation you received assumes an analysis of the company's financials and a public market for their stock, which right now doesn't exist.
Now, that being said...you can take the risk of buying the options and exercising them to own the stock if and only if a) you truly believe the company will (at some point in the future) become a publicly-traded company, and b) you're willing to lose everything you invest if you're wrong. If you're willing to be patient and wait for them to go public, you could be well-rewarded. If not, well...you might be able to wallpaper your house with the stock certificates! (not that anyone prints them anymore, but I remember the days...)
If the money you're going to use to buy the stock isn't money you can't afford to walk away from, I would advise you not to do it without some deep soul-searching about what happens if your choice is wrong.
IF MONEY IS AN ISSUE, AND YOU'RE SO CONCERNED ABOUT RISK, WHY NOT FIND SOMEONE WHO STILL WORKS THERE THAT WOULD BE WILLING TO BUY YOUR OPTIONS FROM YOU?
Check with the company to find out if this is an option, because it might be your best one, and even better, doesn't put your money at risk. You won't make the same kind of money you would if you could exercise them and then sell the stock, but this is "found money" to you anyway.
I'm sure I'll get blowback from some people who'll argue it isn't much money you're putting at risk, but the fact is, it's YOUR money, and only YOU know whether it is money you can afford to lose.
You really have to look at cost/benefit - if you have enough faith in the company and its future prospects to risk $5k+, then you simply have to decide whether you're going to take the leap. But if you do, understand the risks in full before you do it! If you know anyone in the company's accounting department and you can have a conversation with them about it, see if they can shed any light on how the company's doing about paying its bills and getting paid by its customers. Find someone you know in the sales department and see if they'll give you an clues about how the company is doing and what the future looks like. In most large organizations this wouldn't be very easy, but with a small company (and it sounds like this one might still be) you might have some luck.
Whatever you do, again let me emphasize - KNOW THE RISKS AND MAKE SURE YOU'RE READY TO FACE THEM before you do ANYTHING!