Using just try_files didn't work for me - it caused a rewrite or internal redirection cycle error in my logs.
The Nginx docs had some additional details:
http://nginx.org/en/docs/http/ngx_http_core_module.html#try_files
So I ended up using the following:
root /var/www/mysite;
location / {
try_files $uri /base.html;
}
location = /base.html {
expires 30s;
}