Is it possible to serve in multiple branches of the military ?

It is possible but each time you jump you lose seniority in the job and grade. The mode rank you gain the less likely you are to be accepted into another branch without losing rank.

It is possible. I went through tech school in the USAF, with an individual that had already served in the USMC. He finished his contract with the USMC, then he joined the USAF.

I had a good friend of mine. He was a Navy Chaplain serving with a SEAL team in Central America. He continued his college and became a Surgeon and transferred to Delta. He learned some of the things he though changed while working with SEALS. Thus he finished his medical degree. Went to Brag and saw DELTA need folks who could multi task. Thus at his 10 year or so point he went Delta. His words to me were well I can do the job, fix both the internal and mental things that follow the missions. He was not complete until he could participate, and help heal. He is now retired and is a part time minister and a full time fire fighter. To each his own but I suggest you do as he did and master your primary skills.

Re: The comment on Jessica Lynch's book. You're kidding! She got into a truck accideny because the convoy got lost then they were attacked by the enemy. Had no real combat training, broke her legs and was taken to an enemy hospital. Rescued by SF soldiers then had the GALL to accept a Silver Star! PC at its worst! Correct me if I got anything wrong.
She should be an embarrassment to any soldier. I am a Marine vet and I am embarrassed for the army.

Marine Corps 4 years, Air Force 21 years. Retired as a SMSgt

Yes

Of course you can, though as others have pointed out, being flexible and changing AFSC/MOS several times may be necessary. I served 21+ years, including three combat deployments, between 1975 and mandatory age 60 retirement in 2017.
1975-1979 Active Air Force - SIGINT Analyst
1979-1980 Army National Guard -Medium Helicopter repairer
1980-1983 Active Army - SIGINT Analyst
1983-2004 Break in service for College, Defense Contractor
2004-2005 Army National Guard - Flight Operations
2005-2017 Air National Guard - Cyber Systems Operations/Client Systems
2017 Retired from the Air Force as a Master Sergeant

I served in the Army with a great XO who came from the Marine Corps. When asked why he transferred to the Army 82nd. Airborne Division, he stated he wanted to jump out of perfectly good airplanes. The Marine Corps lost a good officer, one of the best I served with. Salute!

i spent 4 1/2 years in the navy as a heavy equipment operator, was in the army as a truck driver and then 2 years in the air force as a nothing, went back into the army.

Well, this guy did four branches... admittedly it was the Reserves, but still, in theory it's feasible to hit a bunch...

https://www.stripes.com/news/20-years-4-branches-servicemember-is-part-of-exclusive-club-1.298471