Step 1: Read both statements.
Statement I: plasmids are autonomously replicating DNA. Statement II: plasmids are extrachromosomal DNA.
Step 2: Recall what plasmids are.
Plasmids are small, circular, double-stranded DNA molecules found in bacteria and some fungi, separate from the main chromosome.
Step 3: Test Statement I.
Plasmids carry their own origin of replication (ori), so they can copy themselves independently of the host chromosome. So Statement I is correct.
Step 4: Test Statement II.
Because they sit outside and apart from the chromosomal DNA, plasmids are indeed extrachromosomal. So Statement II is correct.
Step 5: Note their use.
This self-replicating, extrachromosomal nature is exactly why plasmids are used as vectors in genetic engineering.
Step 6: Conclude.
Both statements are correct, which is option (2).
\[ \boxed{\text{Both Statement I and Statement II are correct}} \]