From what I've heard Star Wars is still using tech from thousands of years prior, and while there are innovations, older craft can still compete with newer craft. So how old is the Millennium Falcon? In the real world you can use a 747 for about 20-40 years (depending on use and the quality of repairs/maintenance). The Falcon seems to be a pile of junk, by the time Han solo has it, other than being extremely fast.
At the time of A New Hope:
Canon - 51+ years
According to the Millennium Falcon IncrediBuilds book, she's been flying the friendly skies for over ninety years by the time of the First Order (emphasis mine):
There are legendary ships, and then there's the Millennium Falcon. For over nine decades, this YT-1300 freighter has sailed across the stars, doing so much more than hauling cargo. Under captains like Lando Calrissian and Han Solo, she's been involved in notorious gambling exchanges and epic smuggling capers. She's outraced kilometer-long star cruisers and holds an unconfirmed record for the Kessel Run. She's even helped topple the Galactic Empire and played a part in winning desperate battles against the First Order.
If we make the assumption that this is dating from the time of The Force Awakens (which seems reasonable), then the Falcon would have been built no later than 24 years before the events of The Phantom Menace1. Since A New Hope takes place 27 years after The Phantom Menace, the Falcon would have been at least 51 years old in that film.
Legends - 60 years
The first chapter of the 2008 novel Millenium Falcon depicts her first day off the assembly line, in the year 60 BBY:
CORELLIAN ENGINEERING CORPORATION ORBITAL ASSEMBLY FACILITY 7
60 YEARS BEFORE THE BATTLE OF YAVIN
Kantt swung back to the transparisteel pane. The droid hadn't moved and was probably continuing to pump fuel into YT 492727ZED.
"That YT," he added when Kantt glanced at him. "A hero's birth if I ever witnessed one. When has that happened?"
Kantt made a face. "It's a freighter, Bon. One of a hundred million."
Bon grinned. "If you ask me, more like one in a hundred million."
Star Wars: Millenium Falcon Chapter 1
So the Falcon is 60 years old in A New Hope.