You can, if you do it properly.
The trick is that a standard Ethernet cable has 8 wires in it, and a standard Ethernet link only needs four of them. This means you can use a special adapter at both ends of the cable, one which connects two ports from your router into the one cable (each port using a different set of four wires) and then another which splits those sets into two separate cables at the other end for connection to your equipment. Note that you will need to use two Ethernet ports on your router (one for your FreeSat and one for your YouView box), despite them being carried over the same length of cable.
You will not be able to do this if you have the thin type of Ethernet cable which only has four wires in it, and you must buy splitter adapters which come as a pair (one for each end) - I have seen so-called "splitters" which were wired in such a way that could not possibly work, and splitters advertised as only needing to be at the far end of the cable - again this cannot work. Buy them as a pair, then you will be able to send them back if they turn out not to work. You will also need four short Ethernet leads, two at each end to connect the equipment to the splitter adapters - so by the time you have bought the splitters and the extra leads, you have to query whether it would actually be no more expensive to add another complete cable.