Do anyone know if military snipers in real life carry any sidearm?
I mean a sniper-rifle cant be that usefull if the enemy gets close.

"Snipers" work as teams. The spotter will have a scoped automatic rifle, plus the other team members pulling security will have automatic rifles and machine guns.

Depends quite a lot on country and branch.

One thing is for sure, is that I've used a military-issue sniper rifle enough to say that only thing in common between a Fallout sniper and a real sniper is the scope. Sniping when standing is almost impossible without a lot of experience, and when crouching it's very difficult (you need to sit to gain more balance). So it should be pretty liberal what you decide to do with your sniper.

Do note that some sniper rifles are equipped with iron sights under the scope for close combat, notably usual service rifles converted into snipers by attaching a scope. However, snipers are not intended to get in messy close quarters fight. In real wars combat is not fought on distances where you see the enemy's eye color - such happens only in very tense urban combat and snipers are not taken into very tense urban combat. They stay behind and do what they can to keep the streets clean.
~ And of course do remember that militaries rarely prohibit bringing your own weaponry to a war.

To answer your question more specifically:

Just say you have a four man team; you'll have the shooter, who has the dedicated sniping rifle, a spotter who is behind a high powered spotting scope to direct fire (he has an automatic rifle too), a rear guard with an automatic rifle + grenade launcher, and then you have a machine gunner with a LMG.

This is professional military. Teams are often bigger nowadays (it used to be two/three, now it's up to 6 or even squad sized).

Pistols are pretty useless in warfare; they're last resort self-defense weapons. With a sniper team, they're not needed as there's so many more effective weapons.

Usually snipers will never get really close. I'm pretty sure they do carry sidearms