Whatever runs under Linux is totally independent of its GPLv2 license. Many, many closed sites run under Linux.
Whatever you write, that only uses Linux' public interface, is yours to license (or not) as you wish. But presumably your software doesn't run directly over the bare kernel, you have to consider the licenses of all the pieces involved. Simplest is to build on some form of prepackaged distribution by a reputable group, where you can rely on them having done whatever diligence is warranted.
Remote use of GPLed programs is allowed without distributing source to users, even if said programs are modified by you from other GPL programs, as you are explicitly allowed to make derivatives for your own use.
Also look for the Affero GPL, which explicitly covers the "use over the 'net" case. A bit more detail on what is or not allowed under different licenses can be found at tldrlegal.