Well presumably you'd exercise at some kind of liquidity event so you can borrow the money for a short period of time. Lots of people will lend you $20k if more than that is coming back soon. Of course if they haven't excited in 7 years then you still have the problem but at least then you'll have a clearer idea of whether the company is going to be valuable and if it's worth excising.
No need to borrow money if the stock is liquid, you can perform a 'sell to cover' transaction, in which your options are exercised and enough of your stock options are sold to pay for that at the same time. Any broker can handle that without a loan required.
