It is possible to plant trees in My Time At Portia, and these will provide your with fruit forever. They take about 14-17 days to fully mature, and will provide regular fruit after they have matured. There are currently 5 different tree seeds and trees in the game; Apple Tree Seed, Apricot Tree Seed, Crystella Tree Seed, Nitra Tree Seed, and Zeolora Tree Seed.
If you want to plant a tree in My Time At Portia, then you need a Large Planter Box. You can buy this box for 2x Data Discs from the Church Store. So if you want multiple trees, prepare to get a lot of Data Discs to buy Large Planter Boxes for.
The tree seeds themselves also cost Data Discs, and also these can be bought at the Church Store. The price varies with the rarity of the tree, and they cost the following:
Apple Tree Seed: 2 Data Discs
Apricot Tree Seed: 3 Data Discs
Crystella Tree Seed: 8 Data Discs
Nitra Tree See: 5 Data Discs
Zeolora Tree Seed: 5 Data Discs
A single tree will essentially set you back at least 4 Data Discs. The more expensive trees will cost a total of 10 Data Discs! Luckily it's not that difficult to come across Data Discs in this game.
When you have a Large Planter Box and a seed, simply plant them like all other seeds in your Workshop. Wait a good while for them to mature, and begin to harvest your fruits after that.