In no particular order I have listed some of the most popular markets for out of state investorsCleveland, OhioCincinnati, OhioDayton, OhioToledo, OhioYoungstown, OhioCincinnati, OhioMemphis, TennesseeBirmingham, AlabamaKansas City, MissouriSaint Louis, MissouriIndianapolis, IndianaDetroit, MichiganErie, PennsylvaniaLouisville, KentuckyMilwaukee, WisconsinJackson, MississippiEach of these markets is popular with turnkey investors because of the low barrier to entry, high rental demand & high rent to price ratio.

The obstacle I'm encountering is that the [conventional] lenders I'm going through keep demanding 20% down (which I don't have---the money I have is being saved for use on rehab deals).

The property is in demand. 

Almost every real estate agent is looking for a great contractor, so you'll be in high demand once you start networking.

At this point in the market cycle even a lot of the historically stagnant markets are experiencing little bumps in value and thinning inventory/higher demand, especially with multi-family buildings.

If you select a market with population growth, jobs and wage growth, a balance between supply and demand and a diverse range of employers you will do just fine.

You want to look in the areas where there is demand for the use you intend to create.

One thing I can think of with buying is you can buy in an established neighborhood that already has rental demand. 

Some will demand 20% of YOUR funds: they want you to have some skin in the game.In summary, commercial lending is as flexible as you and your lender want it to be. 

I'm just an average joe investor compared to many on this site, but I'm an active husband, active father of 4 kids (all 7 and under), active community member, and still work a demanding performance-based full-time W2 job outside of scaling my real estate investments to where they are today.