We are moving and there is no washer or dryer hookups. There is a shed and we wanted to have our washer and gas dryer in it. Is it possible? Also, do we have to have a water heater and hot water hose even if we only was with cold water?

do we have to have a water heater and hot water hose even if we only was with cold water?

No, just cap the hot water in.

Let's look at a few things:

Shed
There can be multiple definitions of a Shed and even locations.
Tell us about the shed.

Power
You need that.

Environment
Freezing - can cause all sorts of issues
Damp - premature rusting
If your in Florida this probably isn't much of a problem.
Critters

Drain
How/where are you going to drain the washer.
How Protecting the trap from freezing

Water
If you really needed to, you could connect using a garden hose from the nearest tap. A vacuum break is a must if doing it this way.

Dryer
Needs power, a vent and a gas line. Might be really difficult. Would suggest electric.

I'm going to ask this me moved to Plumbing.