I need to purchase six slim form factor new desktop towers. Looking for recommendations on a brand known for long lasting high quality components.  Computers aren't used for anything resource intense but they do act as a database for a proprietary program that all the other computers in house talk to. When the computer hosting the database takes a dump, well, the entire show stops.

Had a power supply go bad on a Dell Optiplex 9010, naturally it was on the host computer. Luckily we had a NIB spare and was able to swap the hard drive with minimal downtime.   That was a week ago. Today another one failed at a different location, again show stopper and we are now out of spares. Someone is taking it to a repair shop to see if they can jimmy rig it to work until we can get all new replacements delivered.

Suggestions on what to order? Old spec was Win 10 Pro, i5-3470 3.2ghz with 16 gig ram, 240 gig SSD. They weren't the fastest but worked OK.

You kind of get what you pay for.  The last place I worked, I gave EliteDesk 800 small form factors to our call center.  They were super expensive... but as far as I know, they're still all in use(*), 6 or so years later.

(*)  Except for the ones that went home in employee's purses before they took me seriously about bolting those little things to the desks.  

That being said... if things come to a halt when your DB is down, small form factor desktops aren't what you should be looking at.

Not looking for the super tiny ones, just something slim if possible.   Room could me made for larger towers I'd just have to reconfigure some cabinets

When you try to pack a 24/7/365 heavy duty computer guts into a small compact case you expect to have a higher failure rate of heat generating components like power supplies, motherboards and CPU's than when the same items are housed in a full sized tower with a high wattage power supply and several cooling fans. Hard drives don't like high temperatures.

If you need a server then build or buy a server for the job.

I recently decommissioned a 1990's Pentium II Novell SCSI based server. During it's 30 year life it lost two power supplies. Survived two Mississippi river floods and one massive lighting strike that destroyed all other non computer electrical equipment. It was housed an a full tower case with extra fans and housed in a real server room, AC and heated as needed. Location was in a flood plain and 5 feet above recorded max
historical flood height.

Customer donated old server to us and the case was quite worn & dinged after 30 years use. Re housed old MB in a new tower. Cleaned the drives of all data and used as a backup data for a special Win 98SE and XP network system for a special in house project.

They don't have to run any hotter or last any less time... they'll just make more noise trying to move the same amount of air.

I put some awfully dense blade chassis in place for a company about 10 years ago... they're all still running.  But they sure are loud.  

Go dense.

https://www.supermicro.com/en/products/microcloud