How to store multiple input values with same meta_key

Change your form as suggested:

function startdate() {
    global $post;
    $custom = get_post_custom($post->ID);
    echo "<label>Startdates</label><br/>";
    for ($i=0; $i<count($custom["startdate"]);$i++) {
        echo "<input type=\"text\" name=\"startdate[".$i."]\" value=\"".$custom["startdate"][$i]."\" />";
    }
}

You’ll have to remove and reinstate your individual postmeta entries:

add_action('save_post', 'save_details');

function save_details($post_id) {
    if ($parent_id = wp_is_post_revision($post_id)) $post_id = $parent_id;

    if (!empty($_POST['startdate']) && is_array($_POST['startdate'])) {
        delete_post_meta($post_id, 'startdate');
        foreach ($_POST['startdate'] as $startdate) {
            add_post_meta($post_id, 'startdate', $startdate);
        }
    }
}

Then, of course, you’ll need to add some sort of add/remove mechanism to your metabox form, probably through JS.