There are two RJ11 sockets beside the fibre modem, and also two RJ11 sockets in a room upstairs.
Can we do ethernet?
My hypothesis is that ethernet is possible because RJ45 uses 8 contacts, and 2 RJ11s will have 12 contacts totally (6 each). I would love to hear if there is a practical way to make use of this fact.
RJ11 is smaller, and while it has 6 contacts, only 4 are connected In theory you can do 'fast' ethernet (aka 100mbps) off 4 contacts. With modern devices though there's the possibility of somewhat unexpected behaviour or dropping down to 10 megabit classic ethernet speeds - I've a post about this. For getting ethernet at all, sure, its a simple matter of swapping the keystones, and making sure you have the correct pin outs.
Now the 'interesting' thing here is very often, phone cabling is done with ethernet cable, and its possible you could be running 2 sets of phone jacks off a single cable. In which case converting it to a single ethernet jack makes sense.
Otherwise - while you have the correct number of wires - the twist and order of of the wires inside each cable matter. You could wire up 8 wires to convert it to a single RJ45 but you may potentially face interference issues if its 2 seperate cables. If you have cat5e or 6 cable(s), then just wiring up for RJ45 ethernet should get you what you want.
Probably, but probably not.
As far as I know, RJ11 sockets used for phone lines rarely if ever have all 6 pins wired up; a single phone line uses only a single pair (2 pins), and while 2-line sockets are a thing, the fact that you have two of them suggests they're single-line ones, so you most likely only have 2 pairs total across both of them.
(It is kind of important for Ethernet that the wires come in twisted pairs, with the complementary signals like Tx+ and Tx- always sharing the same pair, so both Ethernet and telephone wiring is generally counted in pairs rather than pins.)
That's actually enough for the older 10/100 Mbps Ethernet (which only needs two pairs and therefore only 4 pins out of 8), but you might have problems with two other issues:
The lower quality cable (phone lines might be Cat5 if you're lucky, but more likely they can be twisted-pair Cat3, and I've also seen non-twisted "twin-lead" cable used for phone) might have issues carrying the higher-frequency Ethernet signals, so you might be stuck with 100 Mbps or even 10 Mbps regardless of the number of pairs available.
(Also, with it being a phone line, even if the cable is Cat5 it wouldn't be surprising to find the installer having left some 10-20 cm of un-twisted wires stashed behind the wall socket.)
The different style of wiring used by telephone lines: it's likely that instead of being point-to-point, the same lines will have been wired across multiple rooms (it used to be common to connect multiple phones to the same line), and that might also cause issues – even if you connect only two Ethernet devices in point-to-point style, a "stub" line going into a 3rd room's RJ11 sockets might cause additional signal degradation.
(I have not tried connecting more than two Ethernet devices to sockets wired in series, but I assume it would fail completely for some devices (as you'd inevitably get Tx-to-Tx and Rx-to-Rx connections when more than 2 are involved) and would probably result in the rest falling back to "half-duplex" Ethernet mode still.)
In fact, regarding point #2, it would also not surprise me if both sockets were wired in series – or if one of them wasn't wired up at all – and there was only a single phone line (1 pair) total...
Instead of raw Ethernet, though, you could try other standards specifically designed for communications over voice-grade lines – for example, VDSL. It seems that VDSL2 point-to-point bridges or "VDSL converters" do exist (such as the PLANET VC-231GF and the now-discontinued Zyxel P-871M) and might give you up to 150 Mbps with just one single-pair RJ11 socket on each end.
There is also a single-pair Ethernet standard, 10BASE-T1, which would support being used over a single Cat3 phone line (with follow-up 100BASE-T1 and so on), but it seems to be more of an industrial thing in practice and I don't know what kind of equipment you would need for that.
You could also find two analog dial-up modems on ebay (the kind that literally communicate over a telephone call), hack together some circuit to provide off-hook voltage, and enjoy a 33.6 kbps PPP connection between them.
