Hello!
While they do not penetrate very far into the body, they can indeed get through.
Commonly, materials that emit alpha, beta or gamma radiation get into the body alone or with other materials which have may become contaminated.
Here are some ways these could get in:
Through breathing; It could be in the air or mixed with dust, which is in the air.
Drinking; It disolves in water, meaning the material could get in your body without you knowing.
In the soil; If soil is contaminated, and plants grow in that soil, you will get a small ammount of alpha and beta particles.
Lastly, animals; Plants and animals can become contaminated, the particles coud be taken in through consumtion.
Hope this helps!
Have a great day!