The first thing I notice is 3 wires. This means likely a battery protection circuit as well. Unknown if it keeps track of charge cycles and just stops working after hitting a max. It's possible that you unwrap the battery, swap the cell, solder the old circuit onto the new cell, but the circuit still tells Drobo that the battery is unhealthy. You might first see if you can see a voltage on the red and black wires. Healthy would be between 3.7 and 4.2 (or as low as 2.6), unhealthy would be over 0 but below 2.6, and if it's 0 on the dot, then that might be the protection circuit cutting off access.
Healthy: battery may or may not be the problem
unhealthy: might be able to get away with swapping the cell
0: maybe should start with getting a new pre-built battery pack
Also, not sure that those batteries emit burnt smells. Admittedly, I don't smell a lot of battery guts, but it seems like every time I do, it's a pleasant, sweet smell, like tutti frutti or bubble gum.
If you can, I'd look for a drobo-specific battery replacement. If it's from Drobo, it likely has a newer model cell but still good quality. It would have battery and whatever circuit already built together. Or if you really want to use it as an excuse to tinker, get a aftermarket replacement battery and replace the cell, so you know that it has a quality cell inside and not some reject or used cell rewrapped for to make a quick $.
Personal experience story: I had an old computer which needed 2x18650 pack. Replaced cells with fresh new matched-pair good ones, but battery still reports dead.
Otherwise:
The cell itself:
Important characteristics:
They don't like to list "maximum discharge current rating" or similar, they do show a 10A voltage curve in the datasheet, so we can maybe assume 10A is the maximum current they recommend to discharge at.
They do recommend charge at 1.370A, so the replacement ought to be able to be charged at that current at least.
The capacity is pretty low, should be easy to find something that has the same or higher capacity.
3.6v nominal, 4.2V max - Basically, bog standard Li-Ion voltages. if you get a higher voltage battery like a 4.35v max, nothing bad, just you won't be able to make use of that last bit of power storage that extra .15V gets you.
I'd say generally, look for a battery which:
Lithium Ion rechargeable
Mentions high discharge current or high power
has a datasheet so that you can see if it allows charging at 1.37A or higher
has a capacity of over 2,000mAh
Flat top (because some places modify them to have button tops, or to have built-on generic protection circuits)
illumn.com has a nice selection of batteries and I trust them to source legitimate batteries. I'd say go with Samsung or Sony, and look for ones with the above traits. (for example, samsung INR18650-30Q, One of the Sony VTC ones, or Sanyo UR18650NSX; ask the shopkeep which is newer, or even which they'd recommend.)
Sources:
I found this related post, theirs has a samsung INR18650-20Q.
https://voltaplex.com/learn/question/need-help-on-replacing-a-battery-for-a-drobo-drive/
Here's some info I found through a 3rd party (panasonic, who bought sanyo's battery division, didn't list a datasheet on their product page) about that sanyo part number:
https://pdf1.alldatasheet.com/datasheet-pdf/view/598217/PANASONICBATTERY/UR18650RX.html
