It’s not impossibe, but it’s unlikely to happen nowadays.
In space there’s no (significant) loss of energy (we’ll ignore tidal forces). Everything has an amount of gravitational potential and kinetic energy relative to everything else. Something that is pulled towards Earth accelerates as it gets closer, turning that gravitational energy to potential energy. Once it’s at the closest point, the object has the same energy still; less gravitational energy, but more kinetic energy (velocity) so will fly off just as far as where it came from. This means with two bodies, there can’t be a change from not orbiting to orbiting. Either the objects came into existence orbiting, they crashed into eachother or they are doomed to fly apart forever.
So if there’s no loss of energy, the only way for something to become trapped in orbit is by transfer of energy. When you have two bodies orbiting a larger body (the sun), then when the two bodies pass by each other from specific directions, it’s possible for some kinetic energy to pass from one body to the other, accelerating one body and decelerating the other. The thing is:
1. There’s a limit as to how much energy can be transferred like this in a single pass
2. Slowing something down requires approaching from a particular direction.
This means the only objects that can realistically be captured by the Earth’s gravity had to start life on a similar orbit to Earth. Most of those have already been captured in orbit – or crashed in to Earth.
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