This question is quite old but I think the answer is still relevant. v11 here. And of course, the web_gantt module is only available in Enterprise Edition so it is not available in Community Edition.

So Odoo uses dhtmlxGantt which is a JS framework. One way to set the tasks (left) column width is to create a new module, and in the JavaScript inherit (include) 'web_gantt.GanttRenderer' and override _configGantt(), call super first and then set this property to a higher value:

gantt.config.grid_width = 250;

Be aware that this will change ALL gantt charts across Odoo. Unless you "extend" instead of "include" in your JS module and have your specific view load your specific JS module bit it this is more complicated and a bit off topic.

See web_gantt/static/src/js/gantt_renderer.js for more info.