Can I play Yu-Gi-Oh! with only level 5 or higher cards?
Is it possible or recommended thing to do?


Possible yes. You can build your deck with the kind of cards you want. If that means a preference for lvl. 5 or higher monsters it's ok, as long as you respect the current banlist and card limits. 

Recommended... perhaps not. Why? Because by only having lvl. 5 or higher monsters the only way to summon them would be by Special Summon or by Tribute Summon... as you have no lvl. 4 or lower monsters that you can Normal Summon without tribute, Tribute summoning your lvl 5+ monsters would be near impossible. 

You would have to rely on strategies revolving cards like Cost Down or Ancient Rules to be able to summon your lvl. 5+ monsters without having to tribute... but that may not be the most efficient strategy to follow (of course, there could be an efficient strategy, but not easy to find).



However, there is an alternative that involves the use of house-rules: Specify the combat rules to be similar to the ones used on Duelist Kingdom Tournament as seen on the Anime. You can select only the ones you prefer, but the one that would solve your query would be, emphasis mine:


  Players could Summon monsters of any Level, without offering Tribute. This meant monsters such as "Dark Magician" and "Blue-Eyes White Dragon" could be Summoned immediately.


Of course, this would not be valid on official games or tournaments, but surely can work out if you are up to consider house rules.