Boots up to error screen suggesting restore/verify/

Actually I should not have said error message - it pops up OS X Utilities with the 4 options: Restore from Time Machine, Reinstall OS X, Get help online, Disk Utility. (1)The first choice was worrying me because I have the two onboard drives and have not established yet if I can address each drive from the TM backup.
(2) If I choose Re-install OS X it gives me only the option to install Mavericks, when I had already upgraded to El Capitan, not having bothered with Yosemite.
(3) I chose to use a Windows PC to access help on line as I did not want to complicate things for the MacMini
(4) Using the Disk Utility option, I can select my larger onboard HDD or my OS X BAse system (Disk 1 as it calls it) and choose Verify Disk not Repair Disk
As you correctly say, it seems you cannot repair the system disk, my SSD, since I had booted from it. However, I can choose Repair Disk for the Seagate Backup Plus Disk, tho why that shoudl be necessary I don't know. Now at the last Mac bootup before getting to the options above, it is looking for Apple Keypad (I use Apple mouse) and Apple wireless keyboard (I use the Apple wired keyboard). Finally it finds one of them and shows the 4 options.
I shall of course follow your advice to look at the link you have given "About OS X Recovery" if I can get to that stage on the MacMini itself. In the meantime, I can only thank you for your response.

I did try the cmd + D startup routine to get internet support - it got me initially to a display of a circle with a diagonal line through it. When I released keys it went to a world globe with maybe a time figure beneath, but that figure changed so slowly that after 15 minutes I gave up and powered off. Trying a supposed "safe boot" using the shift key, did get me to the Apple, after which the OS X Disk Utility screen appeared again. OK so this time I tried to restore from my backup drive that has data (in separate folders) from both the SSD and the onboard second hard drive. For ages it tried to locate a drive to restore to and so I gave up and went to Apple Support Community which is where you find me now - on the MacMini that did indeed come with Mavericks if not the OS before. It had been used as a server machine somewhere in the USA which I guess is why it has two internal drives, the SSD that is of course used for boot and apps, and the 500Gb spinner used for most data. After getting this Mac maybe eighteen months ago, I upgraded the OS eventually to El Capitan.
Yes I do have access to another MacMini as it happens, tho it is having problems of starting a backup and dropping out after a while.

I shall try the copying of files to another machine, via DVD I guess, and I don't yet know which startup files to copy until I look at the link you gave me. I do recall that terminal mode is available from the drop-down menu of OS X disk utility so we shall see ...
Thanks again, Dave

Finally I find the solution - simply SSD failure. I discover from the Apple agent Storefront, that the brand OCZ has a reputation for early failure compared to many top line manufacturers like WD, Seagate, and SanDisk. It is a SanDisk that has been fitted and everything back to normal. Apparently it is better to use the Apple firewire port if at all possible, as it is quicker than USB2, but hey-ho, if it takes an hour to restore data to original place, I am not that bothered.

Thanks you you Illaass, the community keeps faith in Apple.

Cheers, Dave