There is a limit technically, but it's so large, you'll never get anywhere close to it. If you try, then page rendering will get slower and slower until the create/edit screens become useless to the end-user, but to get a measurable slow-down, you are looking at thousands, if not tens of thousands, of entries in the drop-down.
However, your Jira admin is not entirely wrong - you're not an idiot, he's wrong there, but it is rightly his job to question the addition of a drop-down with 250 options.
Your people will struggle to locate the value they want, and the way humans work when presented with more than 20-30 options is to pick the same one they are familiar with, or click a random one.
If they're doing the former, then you should be looking to automate the field so they never have to use it (For example, Dave works for team Rainbow, but only ever enters Violet in the colour field - so set it for him). If they're doing the latter, then you've got too many options and you need to simplify it. Consider using a different type of field (cascading selects are quite handy - they allow you to group options together - maybe Dave could select "Team Rainbow" from the first and then be offered only ROYGBIV as the seven secondary options). 250 options is something I would question in most custom fields. Perfectly valid in a version, especially if you're doing ci/cd, but you can archive versions to keep the list short. But I don't know your use-case, so I can't say that you're wrong, just that your Admin is right to question it.
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