A new set of sails will cost you 70,000 gold pieces. Which in this game is quite a bit of cash. As soon as you purchase your sails the achievement should unlock.
To earn gold in the game just keep completing voyages. Working on a galleon as part of a well organised crew will earn you money faster than working alone. But it's got to be an efficient crew who are focused on making money, not a crew of blood thirsty pirates who chase after every sloop they spot.
If you're good at taking down Skeleton Forts they are the fastest way of earning cash, but be aware that they attract quite a lot of crews so you may end up in prolonged battles.
Edit. You can also unlock this achievement by purchasing a new hull or figure head.
The price for all basic ship upgrades is the same, 70,000 gold. - credit to KawiNinjaRider7.
Further Edit. With the release of the Hungering Deep this achievement has become far easier to achieve. The shipwright now sells basic sails for just 7,000 gold pieces if you've got a little spare cash.
The Hungering Deep quest line also gives you a figurehead when you complete it. - credit to Juif for spotting that one.
Additional edit. Things get even easier. Just buy a flag for 2,500gold and the achievement will pop. - credit to Chameleon1977.