There was never a "town" at Example - it was only a rural post office designation. Example P.O. was about 3 miles south of the Finney county line and about 7 miles west of the Gray county line, in the northeast corner of Haskell county. I believe it would be Section 13, Township 27 South, Range 32 West, although the Section could have varied since the post office was typically located in the private home of the postmaster. Mail was hauled from Santa Fe in the center of the county to the outlying post offices, where the mail was then available to be picked up by residents. Rural delivery routes to individual homes were established much, much later.
(Example was also the name of one of the nine original Haskell county townships - later reconfigured to the three townships that exist today - Dudley, Haskell and Lockport.
Example P.O. was established in June 1886 in what was then Finney county; re-established in what is now Haskell county when that was created in 1887. Example P.O. seems to have been disbanded between about 1890 and 1902, when it was re-established until about 1914 when it was permanently closed, and mail was sent to the Jean P.O., about 6 miles south of Example.
Example does appear on two maps that I'm attaching - one from 1890, the other from 1914.
The Homestead Act became law in 1862, and I think by about 1905 all Kansas lands were under claim. I don't find any Larrabee listed in the GLO land patent database but you're welcome to search -http://www.glorecords.blm.gov/PatentSearch/
If he did not file a claim directly from the government, he could have been a second purchaser, in which case a transaction should be recorded with the Register of Deeds office - you would start with their grantor/grantee indexes -http://www.haskellcounty.org/register.html