No rule of thumb here - but I'll give you my experience both as a broke newbie kid and what I'm doing now :)
When I started out in rentals - I had little cash on hand. My emergency plan was 3 fold...
1. Appeal to older / established family. I did actually use that too! My father in law (who does real estate as well) provided a short term loan when I needed a re-roof. That was super helpful.
2. 401k Loan - This can be a source for a substantial short term loan if required. Of course you are taking your money out of the market and the loan will convert to a taxable and potentially penalized distribution if you leave employment for some reason. So it has it's perils to say the least.
3.. Credit Cards - I've never been particularly spendy. The few times I've relied on plastic for financing where to get thru a real estate crunch. I always made sure to focus on paying them down over a short period.
Now that I've been at this a while, this is what I do, generally...
1. Usually I keep about $20k ready to go. This covers my 5 rental properties, my own personal emergency fund, and a decent size binder in case a deal presents itself (or some other investment). That's probably considered a small amount for the risk of my properties and income as I have a regular job as well. Therefore I also...
2. Invest additional capital in the stock market. I call this 'easy access' money. It's not cash per se... but if you keep too much in cash that's really a wasted opportunity IMO. So I invest the additional - I use a margin account for this purpose. I never borrow for the actual investments, but if I were to need short term access to some of that money, I can temporarily borrow against the account, without realizing any capital gains. This also speeds up the period of time until you can get your money. A stock sale takes 3 days to settle - not a long time, but on a cash foreclosure purchase you don't have 3 days to pay so that won't work!
I guess I can't say I gave you a methodology to tackle this question. Hopefully that gives you some good ideas about the mechanics of handling this predicament though!!!