OS lost on my Note 10.1

The simplest thing to try is a factory reset. I believe that turning off and then pressing and holding power, home and volume up together for a number of seconds is the way you get into recovery on this device. From there you should find options to wipe the cache partition or do a full data wipe (factory reset). I'd try the cache wipe first, because there's a remote chance that will fix it and if it does you've lost nothing. A factory reset will of course lose any data that are not backed-up.

Otherwise Sammobile.com is the place to find official firmware. There may be instructions there too - I think you use a program called Odin on a computer to install it, but as I've never flashed a Samsung I can't explain the procedure myself. If the problem is software of any sort that should fix it.

If the problem is hardware then nothing you do with resets or reflashing software will help.

That sounds like the right mode for the tablet to be in for running Odin. But Odin is an app you run on a computer with the tablet connected via USB.

The "home" button on a Samsung device is the oval-ish one on the front panel. The one that, when you press it, takes you to your home screen.

Your Note 10.1 has different 'modes' it can start up into with 'Recovery Mode' and 'Download Mode' (also referred to as 'Odin Mode' so that's what you're seeing on the screen) being two of them, along with the typical, standard mode you're used to seeing. Recovery and Download are for specific, maintenance purposes that you usually only require for lower level tasks (Recovery being a maintenance mode and Download for those instances where you want to flash a ROM. A ROM is, in a very generalized sense, the operating system.) To manually get into Recovery Mode you'd simultaneously hold down the Volume Up and Power buttons when starting up, and to get into Download Mode you'd simultaneously hold down the Volume Down and Power buttons. (...keep in mind there is no discrete Home button on your model)
Recovery
http://www.hardreset.info/devices/samsung/samsung-n8000-galaxy-note-101-3g/recovery-mode/
Download
http://www.hardreset.info/devices/samsung/samsung-n8000-galaxy-note-101-3g/download-mode/
So if your Note is automatically starting up into only Recovery or Download Modes, that often means something like the Android operating system has become corrupted so your Note can no longer boot up normally. In this instance you'll need to flash a stock ROM to get your Note working again. But if you happen to have your Note in some kind of case, be sure to check it hasn't shifted out of place inside the case, forcing the buttons on the side to be inadvertently pressed down. If you don't have a case though, never mind that suggestion.

To flash a new ROM, you'll need access to a computer. The first thing to try is Samsung's 'Smart Switch' application, this will install the necessary driver on your computer and you can use to try and restore the Android operating system on your Note.
https://www.samsung.com/us/support/owners/app/smart-switch
Hopefully this will do what your need but if not, at least you'll have a working driver installed so you can go through the manual process of flashing the ROM yourself. To do this you'll need to download and install the 'Odin' utility:
http://www.droidviews.com/download-odin-tool-for-samsung-galaxy-devices-all-versions/
Try using the 3.09 version, that's more compatible with KitKat, the version of Android your Note 10.1 is using. The Odin utility is not very backwards compatible so using the latest version might not work out for you. Another issue with the Odin is it's a Windows utility so if you don't have access to a Windows PC, there's also a similar, cross-platform alternative, the Heimdall utility which is available for Linux, Mac, and Windows:
https://glassechidna.com.au/heimdall/
You'll also need to download a stock ROM. Try going here:
https://updato.com/firmware-archive-select-model?q=galaxy+note+10.1&exact=1
and find the ROM that matches your model and your country/carrier. Don't substitute, they're generally not compatible as ROMS are pretty specifically matched to the internal hardware of your particular model. (... If you do use sammobile instead, just keep in mind that the Note 10 firmware is dated enough that it requires you to have a sammobile membership to access the ROMs)
Once you have Odin (or Heimdall) installed and the ROM has finished downloading you're ready to start. At the actual download page of the ROM you've chosen there's an 'Instructions' section that details what steps to take (you'll need to adjust some particulars to match your specific situation -- i.e. don't use the latest version of the Odin utility and to get into Download Mode on your Note 10.1 there is no Home button).