Server side equivalent of this Client Object Model code?

Umm, they are only selecting things on the client side so no, there isn't anything on the server side.

They way i've handled what I think you are trying to accomplish is to pass the values through a querystring. See code below:

function myAction(){
    var items = SP.ListOperation.Selection.getSelectedItems();
    var listId = SP.ListOperation.Selection.getSelectedList();
    var itemCount = CountDictionary(items);

    if (itemCount == 0)
        return;

    var ids = "";
    for (var i = 0; i < itemCount; i++) {
        ids += items[i].id + ";";
    }

    var uri = "_layouts/CustomApp.aspx";
    var queryParams = "?listId=" + listId + "&itemIds=" + ids;
    var relativeUrl = L_Menu_BaseUrl;

    var options = {};
    options.url = appendTrailingSlash(relativeUrl) + uri + queryParams;
    options.title = 'Custom App';
    options.autoSize = true;
    options.showClose = true;
    options.allowMaximize = true;
    options.dialogReturnValueCallback = onFinish;

    SP.UI.ModalDialog.showModalDialog(options);
}


function appendTrailingSlash(url) {
    if (url.length > 0 && url.charAt(url.length - 1) == '/')
        return url;
    else
        return url + '/';
}