They also provide indivisable benefits: the right to vote, and the right to inspect company documents.
The solution to owning "a thousandth of a share" is a stock-split, where the smallest fraction is still one share.
A follow-up question since you know more about this than I do: if some number of people each buy a fraction of a share, who gets to vote that share, nobody? Does it become effectively a non-voting share?
If you take the example of Robinhood, here's the relevant section[1].
> "I understand that fractional shares within My Account (i) are unrecognized, unmarketable, and illiquid outside the Robinhood platform".
[1] https://cdn.robinhood.com/assets/robinhood/legal/Robinhood%2... [Section 27]
However the vote could be exercised by whatever entity actually owns the whole share. That would be entirely a decision for the people who chose to divide up the resource. Presumably BH will only accept a single binary vote per share, so it would be up to the owners of the fractional share to agree (or not).
But: if there was demand, Robinhood could totally offer fractional shares with voting rights.
For each vote, there's typically only a finite number of possibilities.
So Robinhood could just pool the voting intentions for everyone with factional shares, and then vote the whole shares accordingly.
(To be really nice, Robinhood would just own a small handful of extra shares, so that they can round up those pooled votes to full integers.)
Suppose you buy one share of a company that has a thousand shares outstanding. You now own one thousandth of the company, or 1/1000 of the company.
Suppose instead you buy one thousandth of a share. Now you own a thousandth of a thousandth, 1/1000/1000. Or you can call it a millionth of the company, 1/1000000. It's the same number either way you write it.
The point is that a share is already a fraction, and it's no great leap to have a smaller fraction than that.