I'm still exploring and it's getting dark but I can't find a place to rest.

You have to go to the red Workbench and activate it once so that you can then build in Sanctuary. After that go into your preferred house, hold the 'v' key down (not sure what button it is on the consoles) until the construction menu appears. Go to furniture->beds and then select the bed of your choice. If you have the materials you should be able to build it and place it. After that hit Esc to close the construction menu and then use the bed as normal.

Edit: If you don't have the materials then you can select old broken beds and chairs to get wood and cloth and there should be heaps of metal around town too. Highlight the item, press 'r' and then press 'e'.

It looks like I have the material but there's a little man under the required?

Not sure what that is, maybe part of the settlement bed requirements? If there's a green highlight around the bed you can place it.

Npoe. Outline is red. Maybe I'm not far enough along?

You should be able to build stuff immediately after you exit the vault, return home and activate the workbench. Make sure you aren't too close to walls or other items. Try making it in an open area. You can always press 'e' and pick the item up and move it.

You need the first (something?) leader perk under charisma to build beds. There is a bed in the cellar but you don't get the well rested perk.

You don't actually NEED to rest, just fyi.

I definitely don't have any ranks in Local Leader and I was able to make beds. I only have a 5 charisma so I can't take Local Leader, which I plan to, until I get 1 more point in charisma.

You can also rest in the root cellar behind one of the houses, if you're just looking to sleep and heal, or wait out a nuclear rainstorm.

I don't know what you're talking about, I built a bed in the back room of Red Rocket and get the well rested effect after sleeping in it.

OP, if it's outlined red when you try to place it you have to move it to a clear spot. Also you can just sit in a chair to use the wait function to pass time.

I wasn't able to build a bed until I got the local leader perk. I only tried to build the beds with frames. I didn't try to build a mattress bed though.

You don't need Local Leader to build beds.
I was able to lay down a mattress fresh from the vault. Once I had the folks from Concord over I built framed beds. Took Local Leader ten levels later.

You most certainly can build beds without the perk... And large water pumps, turrets, and more. You need that perk for shops.

There is a cellar behind one of the houses which has a bed and few other goodies.

I had the same little red man when I tried to build beds without the local leader perk. Maybe what I (and you) did wrong was try to build a bed in the wrong spot? Meaning try moving the bed around until it turns green and then place it.

Can confirm you don't need any perks to build beds (Xbox One)