The Turtle Boots are Boots available in Golden Sun and Golden Sun: The Lost Age. In Golden Sun they can be found in the overworld west of Kolima. In Golden Sun: The Lost Age they are found in a chest in the Islet Cave. They can be bought for 600 coins and sold for 450 coins.
The Turtle Boots are the worst equippable item in the games, understandable when considering their effect on the wearer's stats. However, they do have a tiny advantage, being the first Boot-class item available in Golden Sun (more boots will not become available until much later).
However, if you want to use the Turtle Boots for added defense, you can lower the cut in agility (but not eliminate it) by equipping the Elven Shirt found in Mogall Forest in Golden Sun. The Elven Shirt increases agility by 1.5x the equipped adepts agility rating while the Turtle Boots lower agility by 0.5. It comes down to personal preference if you decide to use them or not.
In The Lost Age, however, they are strictly a collectible to sell off for money because when it can first be found in the middle of that game, far superior (but still questionable) Safety Boots can be bought much earlier, Safety Boots being like Turtle Boots with more defense and less agility reduction. Even the common Leather Boots you can buy in Madra early on are far better than Turtle Boots because they have more defense and no agility reduction.
One use for them, however, might be slowing down a particular Adept for getting items using the Random Number Generator, as sometimes faster adepts need to be slowed down so they can strike later to make a particular method work.
It should be noted, however that Turtle boots and Safety Boots can both be used to intentionally alter turn order. By guaranteeing the healers move last, this will effectively ensuring that they heal after the round of attacks occur and the other players move. This strategy is best used with a double-healing strategy (i.e. two pure mages), one designated to have high agility increasing items and focusing on fast-healing or stat increases before the turn starts, and the other focusing only on group healing.