Its not much hard to tackle IE8
there are lots of ways to achieve a decent look for IE8
moreover, your website layout is quite simple to get it done.
I see two Issues which are making website look bad in IE8
1) Some html5
tags
2) Some CSS3
Properties
Add below to head tag of your website;
<!--[if lt IE 8]>
<script src="https://oss.maxcdn.com/libs/html5shiv/3.7.0/html5shiv.js"></script>
<script src="https://oss.maxcdn.com/libs/respond.js/1.4.2/respond.min.js"></script>
<![endif]-->
I hope website will look much better in IE8. Moreover, you can add a custom stylesheet for IE8 and can make things good in that;
<!--[if lt IE 8]>
<link href="css/for-ie8-only.css" rel="stylesheet">
<![endif]-->
Last but not the least i recommend to use Modernizer. If you are using html5/css3
.
You can check HTML5 & CSS3 Support (In different browsers) to what is supported in which browser.
A relevant question like yours below.
how to make css3 and html5 compatible website for all browsers including IE7 and later
good luck!