I just kind of think of it as an absolute last resort for emergencies, in the event of a catastrophe (e.g. building burns down).I'm curious how others treat it.

I also have a great contracting company and numbers of plumbers and electricians that have 24-hour availability with Emergency calls. 

As long as a smaller accredited investor has enough reserves, say 3-6 months of emergency money at hand, and doesn't want to lose their hard-earned capital, then cash flowing illiquid assets are a no-brainer to me.

Additionally as a Money Guy listener we are debt free except for the mortgage, I have a 6 month emergency fund saved, and we are saving 25% for retirement.

Its not a dangerous risk if you have appropriate emergency funds for anything that may come up and don't buy a property out of your price range. 

Get contact sheets with emergency contacts for all tenants.

That doesn't mean you shouldn't have easily accessible reserves as well but for the purposes of qualifying for financing and having a diversified investment that can potentially be used in an emergency, I think you should reconsider your view of 401ks. 

A few months later, I received a letter from him, along with an apology and payment for two months rent - he had to return home urgently for a family emergency, bought a plane ticket and left.The "Take me Home" card plays well with our student tenants and their parents (who seem not to get the double entendre of the phase .... or are not willing to acknowledge it ... frankly we missed it initially as well until one of our tenants pointed it out).

Why don't they save up a personal emergency fund?

You can always take out a heloc (especially while you still work) for an opportunity/ emergency fund.