Oh I think Trump is an idiot. 

However, in 2014 the same situation happened and lasted 8 months with Qatar isolated from UAE, Bahrain and Saudi Arabia. The thing is none of these countries is going to start a war. Qatar hosts Al-Udeid airbase, which is the USAF CENTCOM HQ, USCENTCOM Foward HQ, and a sizeable RAF/USAF contingent.
Bahrain hosts NSA Bahrain, HQ for USN 5th Fleet and USN CENTCOM HQ.
UAE hosts USAF and French AF contingents.
Saudi, well it buys it's army from the US, and hosts sizeable US forces.

No one is going to start shooting at one another because the US would be in the middle and would be pissed. Nevermind it would drive Qatar right into the arms of Iran, which is exactly what the Arab League wants to avoid.