A 51 cm for someone 5'10" Sounds small to me, too. But like was said, a lot depends on the geometry of the bike.
Stand flat footed straddling the top tube. For a road bike you should have about 2" clearance between the top tube and your crotch, about 4" for mountain bike. A little more for either is fine.
This measurement is not as important as overall fit.
The only style of bike that you should be easily able to reach the ground with both feet while in the saddle are some of the cruisers like those made by Electra, Rans, etc.