The spell, Tensers Floating Disk states that it is 3ft in diameter and floats 3 ft in the air in an unoccupied space. Since it says that if the stuff its carrying is over its maximum weight, it all falls to the ground, we can assume that it can share its space after it has been cast.
Given this knowledge, and assuming we have a character that can lie under it, covering exactly half of that characters body, can they use it as half cover?
The full description of the disc states:
"...a circular, horizontal plane of force, 3 feet in diameter and 1 inch thick..."
A plane of force doesn't necessarily mean that it's opaque. The closest thing would be the wall of force spell which says that it's invisible. So I would say the disc is also at least translucent if not invisible.
Given that, you could always say the stuff that it's carrying could give you cover. So just drape your open bedroll over it like a tablecloth.
Also, realize that the disc completely horizontal and an inch think. So you'll have to talk to your DM about what "cover" this would provide unless you expect something to come straight down at you.
Speaking of straight down, most all races would need to crawl in order to fit under the three foot height. So for the average humanoid, that would put you in a prone status. Which for ranged weapons, is actually a better defense that half cover; disadvantage for prone averages out to -5 on the to hit roll, whereas half-cover is only a +2 to your AC.
However, being prone means that melee attackers get advantage when attacking you.
So I guess the big question is, what is it you're trying to get cover from?
Adjudicating cover is the task of the DM, so ask your DM how they handle it. However this is unlikely to work against most attackers. The cover rules (p. 196 PHB) state:
A target can benefit from cover only when an attack or other effect originates on the opposite side of the cover. (...) A target has half cover if an obstacle blocks at least half of its body.
The spell creates
a circular, horizontal plane of force, 3 feet in diameter and 1 inch thick, that floats 3 feet above the ground
The pane of force is horizontal. If your attackers are on the ground then their attacks come mostly from the side, and the disc is not a barrier between you and them for most attacks. You are not on the opposite site of it, and it is not blocking at least half of your body. 1
However, against attacks from above, say from a flying manticore's tail spikes, this tactic could work. The DM might consider a large, tall creature like an Ogre also having difficulty to hit you and give cover, also that is a much less clear-cut case.
Depending on the size of your character, you will be in an awkward crouched or lying position, and the DM may impose Disadvantage to counterattacks you undertake, on the grounds that shooting a bow, throwing a javelin or making magical gestures to cast firebolt in such a position constitutes difficult circumstances if crouching, or you would be in the prone condition, if lying, which likewise imposes Disadvantage on your attacks2.
The DM might also rule you are prone if kneeling or crouching, even though prone usually is defined as lying on the ground (p. 190 PHB), as there are no specific rules for how to handle a kneeling or crouching creature, and prone is maybe closer to it than standing. As stated in the opening paragraph, a lot will be up to the DM here to decide.
1 The disk is created in an unoccupied space but nothing later on stops creatures from moving i to its space, or from transporting a creature on the disk, or from sliding under it. It also blocks objects or line of effect, or you could not put anything on it to transport.
2 Prone technically also provides advantage against ranged attacks, which would be yet another benefit on top of cover; however the idea behind this is that you normally present much less of a target to a ranged attacker when lying down. While the rules do not handle it explicitly, that would not really be the case against a flying attacker trying to hit you from above, so be prepared that your DM might rule that prone does not give you this advantage in cases like this.
