Useful information has already been provided by @nathwill on his own answer, however, my considerations are the same as mentioned in the comments to your question, placed here for your convenience:
OwnCloud working pretty fine over here, If you set it on your www
folder simply go to your web browser and try localhost/cloud (change
"cloud" for whatever the name of your cloud folder is), if you used
another method please let us know in order to help you. My Ubuntu Web
Server OwnCloud demo can be reached right here:
geppettvs.servehttp.com/cloud use "guest" as user and "demo" as
password (without quotes, this account will be deleted soon).
OwnCloud is using PHP5 and MySQL services for a web interface, it
should be used by installing it as a web service. Usually placed on
/var/www but these services are mostly installed into their proper
folders which may difficult to find how to call it from the browser. I
simply downloaded the compressed file, unzipped, and placed its
contents on my www folder, deleted the "data" folder inside the
"cloud" folder and called it from a web browser. I named it "cloud" so
you can reach it with that link. I usually reach it via
localhost/cloud and it works.
OwnCloud seems not to be a program but a web service, so you won't
find a way to run it as a program or GUI. I suggest you to give a
chance downloading the compressed file and moving it to your /var/www
folder, then call it via web browser and run the install. 0770
permissions are required in order to run install, MySQL or any
compatible database is also required. After all, everything seems to
be easy and the install runs like a charm. The service is provided
by your own local system and will be online only if your computer is
turned on, bandwidth depends on your ISP connection and the maximum
space that you can use for online storage is defined by the limit of
your hard disk drive or quota, depending what you specify for each
user. The maximum file size that I can upload is 512mb, as mentioned
in the popup when I hover the "upload button". Maybe something is
different in your server/setup which may lead to limit the upload's
filesize.
If you need further information don't hesitate to contact us.
Good luck!