Honestly, unless you can spend more $, one or two USB disks for the mini pc is probably your only choice.

I’m sure plenty will disagree with me, but unless you have specific needs, I’d suggest spending more time sourcing your media rather than rely on transcoding. Most formats of popular stuff are available and Jellyfin will happily play it natively.

Also be aware that transcoding is VERY cpu intensive, unless you have a compatible gpu/transcoder. I run a ML110 with a 6-core Xeon (12 threads) and if Jellyfin needs to transcode something, it uses all of that and still stutters badly when seeking.

If you do need to transcode because you can’t source the media in a compatible way - you may want to use something like Tdarr to transcode it before you try to play it, so it’s ready when you are.

Unfortunately I’m not familiar with the US market, but in Europe we have sites like Geizhals (“Skinflint” in the UK) that are excellent at listing electronics, so you could source them for cheap: internal drives, external drives

If whatever you end up doesn’t have enough 3.5" slots, you can get these things on ebay to use them as a hacked together external housing:

You’d need to get the SATA cables out of your case though.

As for what drives to use: If you don’t need redundancy/parity, then a single 12 (14, 16, 18, whatever you need) TB SATA drive will probably beat everything else pricewise. I’d say that leaves you with roughly $300 for the system itself, if you need to buy a new one.

but doesn’t have any way of adding a bunch of drives

Well, you only really need one or two drives. Are you sure it doesn’t offer any SATA connections?
What about PCIe? You could use a cheap HBA card then.

May I ask: are you sure you need a media center with transcoding? Because it may be totally sufficient for you to access files through a file explorer and play them with VLC/mpv or whatever else. Having a media center is only really useful if you need external access to your media. I set all that stuff up once, then realized i never watch shows/movies on the go. And if I do, i know beforehand and can copy the raw files to the device i plan to watch on.

I pre-processed my library since I know exactly what devices I’ll be using to play my library, so I just found a common format for audio, video, and subtitles.
Then with the help of jellyfin’s transcoding logs I manually created the commands for non-real-time transcoding (depending on your hardware this new command could not make much difference in processing time, but it’ll definitely make a difference in the quality of the output).

With that now I can use rclone to upload my files to a cheap S3 compatible service and just mount it in any computer for jellyfin to read.
I’m using iDrive e2. Although I’m only using a few TB while I buy a HDD, so for a dozen or more, I’ll definitely will go directly to buy my own hardware.

For drives, Shuckstop has a table of current shuckable drive prices. Shucking is usually the cheapest way to get new drives, you just have to get them out of their external case (or in your case, leave them in and plug them into your pi, ideally with some sort of fan). There’s also ServerPartDeals for refurbished enterprise drives.

Perhaps you could be interested in Tdarr, so you can transcode your media files to formats you can directly consume in your different devices.

That could save you the need to buy new hardware for on-the-fly work, and also give you control about the metadata, audio tracks and subs included in the final file.

Personally I run 1 NUC with a 8TB USB-HDD and mirror it once a week to my other Pi4 with the same HDD.
Those were kinda cheap at 150€ each but the performance isnt very great.

Check out the Asustor AS5202T

You could get it and a pair of 8TB HDDs for around your $500 mark. Or spring for the 4 bay model and get cheaper drives.

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https://serverpartdeals.com/collections/manufacturer-recertified-drives/products/seagate-exos-x18-st16000nm000j-16tb-7-2k-rpm-sata-6gb-s-3-5-recertified-hard-drive