Messi needs to turn this around ASAP, incredible players such as Van Basten and Ronaldo were also never able to play their careers to the fullest due to injuries. Lack of rest is not the issue, he's been played far more often before and still not even gotten tired, let alone injured. Hell for a guy his age, compromised muscular durability is more likely to be attributed to lack of endurance training than lack of rest. Messi's often either walking or sprinting these days; his incredibly bulky legs coupled with his explosive and technical playing style could suggest that most of his entire routine focuses on strength/weights exercises or technique. He'd probably have far less injury troubles if he gave a larger amount of focus to endurance based cardio training like what they do for triathlons. It'd build stamina and muscular endurance, both of which he's been lacking for quite a while now.
Not knowing everything about Messi ensures that anything we say here is pure speculation but my theory is that he's been inconsistent with his fitness regime for the past 6 months and hasn't kept steady on his routine. Every month or so there's always some article about Messi beginning a new "special diet" or having a new personal trainer or something changed in his regimen. If he finds the right physical system for him, sticks to it religiously, and has unbelievably keen and meticulous fitness professional watching over him all the time then he could easily fix his injury issues.