I learned to drive in a 1990 Peugeot which quite commonly did this.
First off, as there is no grinding of gears, it is likely that the cogs in the gearbox are not turning when you are trying to engage them.
The second thing to keep in mind is that you are talking about automotive technology which is now over 30 years old and therefore doesn't have as much of the assistive features of more modern cars.
What used to happen with my old Peugeot was that gear cogs would meet each other unevenly, that is tooth-to-tooth or gap-to-gap rather than nicely tooth-to-gap. First and reverse were always the gears which it would happen with most often as they are the lowest ratio gears in the box.
So what can you do? Well my solution was to press the clutch pedal all the way to the floor and let it all the way back up, then do this again before then pushing the clutch down one final time and engaging the gear. This seemed to nudge the cog around enough to allow the gear to then engage.
This may or may not work for you, and it may or may not be the actual reason (I would say there isn't enough info for a definitive diagnosis) but given as all other answers involve some work and money, and this you can try with no risk to yourself or your machine, then why not give it a go? Let me know how you get on!