Best way of aggregating, storing and using data in SQL Server (triggers, scheduled jobs, SSAS?)
The aggregate table can probably stay in the same DB. You can however create it on a separate filegroup and disk.

Full update of a new aggregate table

One way of doing it using pure SQL and your sample:

WITH list ([Type], [FeatureId], [Measure], [Value], [Timestamp], [ID]) as (
    SELECT *
        , ROW_NUMBER() OVER(PARTITION BY [FeatureId] ORDER BY [Measure]) 
    FROM (
        SELECT 0, [FeatureId], [MeasureFrom], [Value], [Timestamp] FROM data
        UNION ALL
        SELECT 1, [FeatureId], [MeasureTo], [Value], [Timestamp] FROM data
    ) l([Type], [FeatureId], [Measure], [Value], [Timestamp])
)
SELECT l1.FeatureId, [MeasureFrom] = l1.Measure, [MeasureTo] = l2.Measure
    , [Value] = CASE WHEN l1.Type = 0 THEN l1.Value ELSE l2.Value END
    , [Timestamp] = CASE WHEN l1.Type = 0 THEN l1.Timestamp ELSE l2.Timestamp END
FROM list l1
INNER JOIN list l2 ON l1.FeatureId = l2.FeatureId AND l1.ID+1 = l2.ID
;


With this really small sample, I am not sure it covers all your needs. It may help to add a bigger sample with more data.

It does 4 table scans. Since you have 250k rows, it may not perform so well. It will probablibly better to run it in batch of X consecutive FeatureIds. 

This could be done using a job and a SSIS package with either a single update or batch update. You would have to truncate the aggregate table first.

Trigger on each new row in main table

For new rows, using a trigger, this query could be used:

WITH new([FeatureId], [MeasureFrom], [MeasureTo], [Value], [Timestamp]) as (
--    SELECT 1, 1, 20, 2, '2015-01-01'
--  SELECT 1, 5, 15, 3, '2015-01-02'
    SELECT 1, 9, 10, 8, '2015-01-03'
), gap([FeatureId], mn, mx) as (
    SELECT n.[FeatureId], mn.mn, mx.mx
    FROM new n
    CROSS APPLY (SELECT mn = MAX([MeasureFrom]) FROM data3 WHERE [FeatureId] = n.[FeatureId] AND [MeasureFrom] &lt; n.MeasureFrom) mn
    CROSS APPLY (SELECT mx = MIN([MeasureTo]) FROM data3 WHERE [FeatureId] = n.[FeatureId] AND [MeasureTo] &gt; n.MeasureTo) mx
), list (x,[FeatureId], [Measure], [Value], [Timestamp], [ID]) as (
    SELECT *
        , ROW_NUMBER() OVER(PARTITION BY [FeatureId] ORDER BY [Measure]) 
    FROM (
        SELECT 0, d.[FeatureId], d.[MeasureFrom], d.[Value], d.[Timestamp] 
        FROM data3 d
        INNER JOIN gap g ON d.[FeatureId] = g.[FeatureId] AND (d.MeasureFrom &gt;= g.mn AND d.MeasureFrom &lt; g.mx)
        UNION ALL
        SELECT 1, d.[FeatureId], d.[MeasureTo], d.[Value], d.[Timestamp] 
        FROM data3 d
        INNER JOIN gap g ON d.[FeatureId] = g.[FeatureId] AND d.MeasureTo = g.mx
        UNION ALL
        SELECT 2, [FeatureId], [MeasureFrom], [Value], [Timestamp] FROM new
        UNION ALL
        SELECT 3, [FeatureId], [MeasureTo], [Value], [Timestamp] FROM new
    ) l(x, [FeatureId], [Measure], [Value], [Timestamp])
) 
MERGE data3 AS target
USING (
    SELECT l1.FeatureId, [MeasureFrom] = l1.Measure, [MeasureTo] = l2.Measure
        , [Value] = COALESCE(d.[Value], l1.[Value])
        , [Timestamp] = COALESCE(d.[Timestamp], l1.[Timestamp])
    FROM list l1
    INNER JOIN list l2 ON l1.FeatureId = l2.FeatureId AND l1.ID+1 = l2.ID
    LEFT JOIN data3 d ON d.MeasureFrom = l1.Measure OR d.MeasureTo = l2.Measure
) as source(FeatureId, [MeasureFrom], [MeasureTo], [Value], [Timestamp])
ON (target.FeatureId = source.FeatureId AND target.[MeasureFrom] = source.[MeasureFrom])
WHEN MATCHED THEN
    UPDATE SET target.[MeasureTo] = source.[MeasureTo]
WHEN NOT MATCHED BY target THEN
    INSERT (FeatureId, [MeasureFrom], [MeasureTo], [Value], [Timestamp])
    VALUES (source.FeatureId, source.[MeasureFrom], source.[MeasureTo], source.[Value], source.[Timestamp])
;


You must replace the new CTE by the values of the new inserted row in the trigger and use it to merge with the aggregate table.

Scheduled update of newly added rows

If you can find a way to get a list of all newly added rows to the main table since the last update of the aggregate table, you could schedule a job every x minutes or hours and only update what is necessary based on what has been recently added.

The trigger query will work as well with scheduled updates.

This can be run as a schedueled job and within a SSIS package.