Welcome new user.

Yes, the shares are safely yours forever.

However...

It may be you're thinking about "takeovers" or obscure situations such as public companies going private.

If for example, incredibly, some group of people "privatized" Apple, there would be a vote on it. (You'd vote as one of the shareholders). If the "yes" vote won, it would not be a public company anymore, and indeed, you would get a pile of money for your shares.

So just to be clear, it's quite impossible for anyone to "steal" or "just take" your shares. But yes in obscure situations, say (in short) someone "bought all of Apple totally" then (of course) you would get a pile of money for your shares.

Another example is, if Apple went completely bankrupt, whatever amount of money was left over, you would get your share.

In short, in the sense you are asking the question, you're safe, nobody can ever take away your shares.

(Sure, if the "whole company was sold" or "went bankrupt", which is very obscure, all shares would cease to exist, but, you would certainly get your share of the money.)

The only way I can think someone could literally "take away your shares" is if, there was a political change and the government stole Apple. This happens from time to time in history. Some governments will decide that - for example - a railroad or coal mine should belong to the government, so they just take it.

So to directly answer...

Do I get to keep these shares forever?

Yes.

Is there any chance that the stock exchange could use some powers of eminent domain to take away my shares and pay me a small compensation?

Yes:

in the incredibly unusual case that Apple "goes private". But note that you would get a LOT of money in that case! (It's easy to google articles on this ... article . ) Note that you, as a shareholder, would indeed be one of the people voting on whether or not to sell Apple "as a whole".

Consider a simple "restaurant example" ...

This is much like owning a company with friends. Imagine 10 of you own a restaurant, and you have put in $100,000 each. Say the restaurant is going well. Someone offers you all $5m to buy the restaurant. (ie, $500,000 each.) It could be that the 10 of you all agreed at the beginning, you will never sell it, you're not allowed to sell it. In that case, obviously, it won't be sold. However, it could be that you agreed that if 6+ of you vote to sell it, then it is OK to sell it. In that case even if you voted No, it would be sold (and of course you'd get your $500,000).

Public companies are (generally) like the latter case in the restaurant example. They can be "sold out" if there's a vote Yes to sell out - but don't forget this is incredibly unlikely. I'm just giving you the "total answer".

The fact is yes, the shares can be taken from you (just as in the "restaurant example", but you will have voted and generally you get a pile of money.

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One detail. You probably know that shares sometimes "split". All this means is, for every one share (trading at say $80) you get two instead (each trading at say $40). This is completely common and normal. In a sense technically they "take away" your old shares and give you the 2x new ones. So sure, in a sense that is an example of the system "taking" your shares forcibly, you don't have any say in the matter.