In the latter half of the 19th century, my great grandfather wrote a series of handwritten poems that he compiled in handbound poetry book, along with two of his penciled drawings. The book was never submitted for publication. Can I copyright this book, and then submit for publication?
When your great grandfather committed the poetry to paper, he started the clock on his copyright. It runs out -- for most cases -- 70 years after his death. Assuming he has passed, then the copyright transferred to his estate and transfers to his heirs.
If your great grandfather was a corporate entity -- like Walt Disney -- then the copyright period is 120 years from the creation of the work.
Even if the copyright has expired, you can still publish the work, just without protection. Ownership of the literary rights, and who gets the money would be thorny absent specific declarations in the wills of great grandfather and successive generations.
While the original work is your personal property, the literary rights - and subsequence revenue - could belong to every living descendent. Of course, zero split a thousand ways is still zero, so there may not be anything stopping you from self publishing the work so the family can see their great grandfathers work in print.
While the copyright has been harmonised internationally, certain aspects are regulated differently in each country. One of those aspects is the duration of protection. You have not told us where you live or where you want to publish the work, so all I can do is to link to a list where you can lookup the concrete length or protection for your country: List of countries' copyright lengths. You will find durations based on the author's death and based on creation/publication date. So to answer the question, if the work is still protected under those rules, you can use copyright protection.
Assuming the poems are now technically in the public domain, you could do what most people do in this situation --add an introduction, some biographical information, some essays, and some new original poems (all copyrightable material) and then submit that for publication.
Your grandfather's poems will remain be in the public domain, but the rest of the book would be under your copyright.
It's worth noting, however, that poetry is difficult to sell under the best of circumstances. Unless your grandfather was an extraordinary poet (and/or a figure of historical importance), your best bet of seeing this in print is to self-publish, and with no expectation of recouping your costs.