A key part of Meld into Stone, as Groody the Hobgoblin mentioned, is that you “can't move.”
You can use your movement to leave the stone where you entered it, which ends the spell. You otherwise can't move.
This question discusses the difference between “can't move” and “movement speed becomes zero.” The currently accepted answer is that they both just set your movement speed to zero, but “can't move” has the additional effect that changes to your base speed have no effect. What's important here is that, even being unable to take movement (the game term), you can still move. Otherwise (as that answer mentions) you'd have a condition like Incapacitated or Paralyzed. You just can't leave your spot.
So, under the effects of Meld into Stone, you should be able to do things that any other character with no movement speed can do, such as:
- swing your arm, as if wielding a weapon;
- make somatic gestures to cast spells;
- squirm, kick, and wriggle, like someone attempting to escape a Grapple; or
- fall prone (“You can drop prone without using any of your speed.” (Basic Rules, Ch 9, “Being Prone”)).
So you could potentially just fall down prone and sleep on the ground — assuming that would be a reasonable (if uncomfortable) way to sleep were you not melded into stone. You could even change positions in your sleep; you just wouldn't be able to roll away from your spot. (Or have a midnight snack or some water, since your equipment is also melded into the stone.)
The Most Comfortable Sleep You've Ever Had
Note the following effect of the spell:
Nothing of your presence remains visible or otherwise detectable by nonmagical senses.
You, the caster, are not explicitly excluded from this effect. It's entirely possible that, while under the effect of Meld into Stone, you are not able to perceive your physical self at all — a perfect isolation chamber. So you may not even have to lie down, because you won't be able to feel your feet or legs. You may just drift in a peaceful absence of physical stimuli, in complete darkness, only vaguely hearing the sounds around you.
It could be incredibly peaceful. Or the loss of stimulus could lead to hallucinations and complete disorientation.