A 10 gallon tank is the minimum to keep betta with other fish, but if you are doing neons I would go up to a 15 or 20 gal (longer the better). Neons have a nippy reputation, but this can be fixed by keeping them in groups of 10 or more. Even in a group of 6 they will still be nippy.
Just as a general fishkeeping note, I don't recommend neon tetra at all. They need rather soft water to thrive and show their best colors. Due to inbreeding from production on a commercial scale, they are also rather weak and sickly. X-ray tetra (Pristella tetra), harlequin rasbora, and glowlight tetra make much better tankmates due to their adaptability to many different water supplies and their peacefulness. But all of these fish still need a minimum of 6. I recommend starting with 9 for better results that way in case you lose one, all hell won't break loose in the tank.