I think something can exist by itself. Indeed, I think some things must be capable of this, otherwise nothing whatever could exist (and clearly some things exist).
Every object that exists is either complex or simple. That is, it is either made of simpler things (in which case it is complex) or it is made of nothing simpler than itself. Those exhaust the possibilities. Complex and simple - everything existent is one or the other.
Clearly not everything can be complex. For that would lead to an infinity of complex things and there cannot be an infinity of anything.
So, if there are some complex things, then there must be some simple things from which the complex things are constructed.
All extended things - so all objects that occupy space - are complex. For all objects possessing extension can be divided. Thus, no extended object is a simple object.
A simple thing, then, must lack extension. That is, it is must be something that does not occupy space (for if it occupied any space, it could be divided and that would demonstrate it to be complex).
The simple things - things that must exist if anything exists - therefore do not occupy any space.
Furthermore, simple things, by their very nature, exist. For a simple thing can neither be constructed - for there is nothing from which one can construct one, as they have no ingredients apart from themselves - not destroyed (for there is nothing into which one can deconstruct it). Thus simple things exist, and exist of necessity.
Complex things cannot exist absent simple things, but simple things can exist absent complex things.
So, can anything exist by itself - yes, a simple thing can.
Do such things exist - yes, demonstrably.