A lot of people that come into Archeage ask this question regularly and aren't really sure whether its possible. I have done some guides on how to make patron and while there's always the option that a free to play player can simply keep a character on all night long to regain labor there are some disadvantages.
One of the first disadvantages is owning land. When you enter the game you get a 8x8 scarecrow for doing some of the very first quests around level 10. You cannot use it without patron even though it's giving you taxes and the design. So this kinda tricks the person into thinking they can place it and find out later it can't be placed without patron.
The next thing is everything in game is powered by labor. During alpha you had some fun things like treasure hunting which didn't cost labor. But now the only thing that you can do is dungeons or farming monsters but if you find any gear or try to open coin purses this will cost labor. The game is powered by labor crafting and gathering is labor intensive. Which makes the grind that much harder for free to play players. The only things you can do to make money which cost absolutly no labor is farming certain designs but these are high competition.
Making enough money to actually get patron is still possible but trying to play completely free you won't get super far unless you work really hard. This doesn't mean that the game isn't playable. The game isn't a pay to unlock like SWTOR or doesn't force you to pay a subscription to play the game. And while people might disagree the game isn't pay to win. Yes if you dump enough real money into any game there will be some kind of advantage whether its decreasing time it takes to get things but it's impossible to get away completely from these types of things. If it wasn't for the shop the gold farmers would sell the gear which would be much of the same thing.
But that doesn't mean that some things are broken about the game. One being the RNG boxes which are sold in game. In Korea you can buy most of the regrade items for a set number of credits. With the NA servers they have it set so that if you spend 100 dollars the chance of you getting the item you want is still very low. Which means regrades which make a major difference at end game are only for the high rollers. The reason is because they can't be bought for low credit amount we have to rely on a "luck" chance at said items.
The game could be balanced out farther which would make it a little bit more fair to players some things include adding better incentives for patrons such as increasing the labor increase to alpha labor which was 20 per 10 minutes instead of only 10. Right now since the gold farmers can sell gold at a better deal then buying apex for gold the cost of labor has increased so much that normal non pay to play characters are unable to buy labor and make money with said labor. The labor potions only costed 150 credits when first released but cost 300 right now. Its not worth it for people to buy and sell these either. The only labor potions are the ones bought by loyalty that people still sell. This makes it difficult to obtain them aside from people spending loyalty to sell them.
The people who buy gold from gold spammers will also control the markets on various items which also helps ruin the game. But if trion makes it more worth selling apex then it is to buy gold from gold spammers they will take the game back and recover many things.
At the end of the day I would have to say out of all the other older free to play mmorpgs that actually have decent graphic this is one of the better ones. You have people still play nexon and Conquer Online and many other hardcore pay to win games which even lack the good quality graphics. Archeage has really good pvp and will likely continue to be popular since it's one of the ages for more hardcore pvp that we have seen for a long time.