The most common and most likely answer to this crossword clue is the 4 letter word NAIL
I've seen this crossword clue in The Sunday Telegraph
N
A
I
L
More clues leading to the result NAIL
Buddy explains!

The straight part
We know that "very anxious" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case very anxious = nail.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does NAIL mean?
NAIL noun- The thin, horny plate at the ends of fingers and toes on humans and some other animals.
- The basal thickened portion of the anterior wings of certain hemiptera.
- The terminal horny plate on the beak of ducks, and other allied birds.
- The claw of a bird or other animal.
- A spike-shaped metal fastener used for joining wood or similar materials. The nail is generally driven through two or more layers of material by means of impacts from a hammer or other device. It is then held in place by friction.
- A round pedestal on which merchants once carried out their business, such as the four nails outside The Exchange, Bristol.
- An archaic English unit of length equivalent to 1/20 of an ell or 1/16 of a yard (2 1/4 inches or 5.715 cm).
- To fix (an object) to another object using a nail.
- To drive a nail.
- To stud or boss with nails, or as if with nails.
- To catch.
- To expose as a sham.
- To accomplish (a task) completely and successfully.
- To hit (a target) effectively with some weapon.
- Of a male, to engage in sexual intercourse with.
- To spike, as a cannon.
- To nail down: to make certain, or confirm.