"Hey! I gotta idea for a time travel adventure!"
"What hellhole are you... I mean, what hellhole is TimeWatch sending us to?"
"How about a relaxing trip to the Mediterranean?"
"How about telling us what the catch is?"
The Messinian Salinity Crisis
The historical significance of the Mediterranean Sea and the area surrounding it is undeniable, but that particular body of water wasn't always there. The Mediterranean Sea dried up a little less than 6 million years back in an event called the Messinian salinity crisis. The Straits of Gibraltar closed off the Mediterranean from the Atlantic Ocean. The hot and dry climate did the rest through evaporation, although lakes of brackish water remained.
Anybody traveling to the area during this period could encounter a deep dry basin. The dry climate and higher air pressures due to being up to 3-5km below sea level combined with summer heat to create desert like conditions. Much of the area would be covered in a layer of salt.
Why would anybody take a time machine to such a place? It makes for a good location for a temporary base. The inhospitable climate means that nobody and nothing is likely to just wander by and be a bother for a period of a few hundred thousand years. Evidence of the base's existence will be power washed away when the Mediterranean Basin becomes the Mediterranean Sea again (see below). Of course, it being such prime real estate for a time traveler in the market for a secure and isolated base location means that it might also be one of the first places for people to look.
A hydroelectric dam placed between the Mediterranean Basin and the Atlantic Ocean could be a vast source of electrical power. It could power a base or be the whole motive for going there. The idea of damming off the Mediterranean Sea came up in the 1920s as the Atlantropa proposal. Of course, methods of time or dimensional travel that use more power than such a dam would produce would make the whole venture unprofitable.
The Zanclean Flood
The Messinian salinity crisis ended about 5.33 million years ago with the Zanclean flood. The triggering event was the formation of the Straits of Gibraltar. It took anywhere between several months and two years for most of the Mediterranean Sea to refill. Signs of erosion from the massive flow of water from the Atlantic Ocean suggest that flow rates were impressive.
Given the historical significance of the Mediterranean, preventing the Zanclean flood is of interest to groups with an eye towards altering human history. However, an effort on that scale would be difficult to hide from anybody with the ability to monitor for historical changes. This might lead to such a project being used to conceal or distract from more subtle time travel shenanigans.
Going back to make sure the Zanclean flood happens on time could involve anything from being issued shovels as a punishment detail to the faster and more fun method of explosives. The Zanclean flood could be deliberately triggered to destroy a decommissioned base in the Mediterranean Basin. It could also be used to destroy an effort to make the Atlantropa proposal happen during this period.
Finally, the Zanclean flood might have been an accident. An example of collateral damage from an exchange of eldritch weapons during a time war. Or the result of teleporting a nuke into the past to stop it going off someplace worse.