The most common and most likely answer to this crossword clue is the 8 letter word OFF-LINE
I've seen this crossword clue in The Times
O
F
F
-
L
I
N
E
More clues leading to the result OFF-LINE
Buddy explains!

The straight part
We know that "Avoiding the web?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case avoiding the web? = off-line.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 OFF-LINE mean?
OFF-LINE adjective- Of a system, currently not connected (generally electrically) to a larger network. For example, a power plant which is not connected to the grid, or a computer which is not connected to the Internet or to any other communications service.
- (by extension) Happening in the physical world (the real world) as opposed to on the internet.