On Linux with GNU grep:
grep -F --text -o --byte-offset mystring binaryfile
$ grep -F --text -o --byte-offset option /bin/tar
Explanation of the parameters:
Interpret PATTERN as a list of fixed strings, separated by newlines,
any of which is to be matched. (-F is specified by POSIX.)
Process a binary file as if it were text; this is equivalent to the
Print only the matched (non-empty) parts of a matching line, with
each such part on a separate output line.
Print the 0-based byte offset within the input file before each line
of output. If -o (--only-matching) is specified, print the offset of
the matching part itself
To count the occurrences add
| wc -l to the command line.